Benjamin Bona Mahony Equation with Variable Coefficients: Conservation Laws
|
|
- Audrey Stevens
- 5 years ago
- Views:
Transcription
1 Symmetry 2014, 6, ; doi: /sym OPEN ACCESS symmetry ISSN Article Benjamin Bona Mahony Equation with Variable Coefficients: Conservation Laws Ben Muatjetjeja and Chaudry Masood Khalique * International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa; Ben.Muatjetjeja@nwu.ac.za * Author to whom correspondence should be addressed; Masood.Khalique@nwu.ac.za; Tel: ; Fax: External Editor: Slavik Jablan Received: 22 August 2014; in revised form: 18 November 2014 / Accepted: 25 November 2014 / Published: 15 December 2014 Abstract: This paper aims to construct conservation laws for a Benjamin Bona Mahony equation with variable coefficients, which is a third-order partial differential equation. This equation does not have a Lagrangian and so we transform it to a fourth-order partial differential equation, which has a Lagrangian. The Noether approach is then employed to construct the conservation laws. It so happens that the derived conserved quantities fail to satisfy the divergence criterion and so one needs to make adjustments to the derived conserved quantities in order to satisfy the divergence condition. The conservation laws are then expressed in the original variable. Finally, a conservation law is used to obtain exact solution of a special case of the Benjamin Bona Mahony equation. Keywords: Benjamin Bona Mahony equation with variable coefficients; lagrangian; noether operators; conservation laws 1. Introduction The Benjamin Bona Mahony equation u t + u x + uu x u xxt = 0 (1)
2 Symmetry 2014, was, for the first time, studied by Benjamin et al. [1]. It is also known as the regularized long-wave equation and is applicable to shallow water waves and to the study of drift waves in plasma or the Rossby waves in rotating fluids. See, for example, [2] and references therein. It should be noted, however, that the BBM equation was also derived by Peregrine [3]. In the last two decades, various versions of the Benjamin Bona Mahony equation have been investigated in the literature. A general form of the Benjamin Bona Mahony equation is [2] u t + αu x + βuu x + δu xxt = 0 (2) where α, β and δ are constants with the nonlinear and dispersion coefficients β 0 and δ < 0, respectively. For different values of the constants α, β and δ, Equation (2), results in various types of nonlinear equations which are very useful in the study of various physical phenomena. See [2] and references therein. However, in certain cases, the physical situations of the problem dictate us to consider nonlinear equations with variable coefficients [2,4 8]. Recently, in [2] the variable coefficients version of the Benjamin Bona Mahony equation u t + α(t)u x + β(t)uu x δ(t)u xxt = 0 (3) was investigated and some exact solutions were obtained using the classical Lie group method [9]. In this study, we consider the Equation (3), but with δ being an arbitrary constant, namely u t + α(t)u x + β(t)uu x + δu xxt = 0 (4) where α(t) and β(t) are arbitrary functions of t. The objective of the study is to classify the Noether operators and to construct conservation laws for the Equation (4). Conservation laws are mathematical expressions of the physical laws, such as conservation of energy, mass, momentum and so on. They play a very important role in the solution and reduction of partial differential equations. Conservation laws have been widely used to investigate the existence, uniqueness and stability of solutions of nonlinear partial differential equations. This can be seen in the references [10 12]. They have also been employed in the development and use of numerical methods (see for example, [13,14]). Recently, conserved vectors associated with Lie point symmetries have been used to find exact solutions of some partial differential equations [15] and systems of ordinary equations [16]. Noether theorem [17] gives us an elegant way to construct conservation laws provided a Lagrangian is known for an Euler Lagrange equation. Thus, the knowledge of a Lagrangian is essential in this case. It is worth mentioning that no Lagrangian exists for Equation (4) and as a result one can not invoke Noether theorem. However, an interesting approach is employed to construct conservation laws for Equation (4). It should be noted that the present approach, which we use here, fails to construct the conservation laws for Equation (3), i.e., when δ is a function of t. The paper is organized as follows. In Section 2 we briefly give the fundamental notations and relations concerning the Noether symmetry approach. Section 3 obtains the Noether operators and the corresponding conservation laws for the Equation (4). In Section 4, a conservation law is used to obtain exact solution of a special case of the Benjamin Bona Mahony equation. Concluding remarks are presented in Section 5.
3 Symmetry 2014, Fundamental Notations and Relations Here we present some vital features of Noether operators concerning partial differential equations. These results will be used in Section 3. The reader is referred to [17,18] for further details. Consider the vector field X = τ(t, x, z) t which has the second-order prolongation X [2] given by + ξ(t, x, z) x + η(t, x, z) z X [2] = τ(t, x, z) + ξ(t, x, z) t x + η(t, x, z) z + ζ1 t + z t ζx 2 + ζ 1 tt + ζ 2 xx + ζ 1 tx + (6) z x z tt z xx z tx where the expressions for ζ 1 t, ζ 2 x, ζ 1 tx and ζ 2 xx are given in [5]. The Euler Lagrange operator is defined by (5) δ δz = z D t D x + D 2 t + D 2 x + D x D t (7) z t z x z tt z xx z xt where the total differential operators are given by D t = t + z t z + z tt + z xt + z t z x (8) D x = x + z x z + z xx + z xt + z x z t (9) Consider a partial differential equation of two independent variables, viz., E(t, x, z, z x, z t, z tt, z xx, ) = 0 (10) which has a second-order Lagrangian L, i.e., Equation (10) is equivalent to the Euler Lagrange equation δl δz = 0 (11) Definition 1. The vector field X, of the form Equation (5), is called a Noether operator corresponding to a second-order Lagrangian L of Equation (10) if X [2] (L) + {D t (ξ 1 ) + D x (ξ 2 )}L = D t (B 1 ) + D x (B 2 ) (12) for some gauge functions B 1 (t, x, z) and B 2 (t, x, z). We now recall the following theorem.
4 Symmetry 2014, Theorem 1. (Noether [17]) If X, as given in Equation (5), is a Noether point symmetry generator corresponding to a second-order Lagrangian L of Equation (10), then the vector T = (T 1, T 2 ) with components T 1 = τl + W δl δz t + D x (W ) δl δz tx + D t (W ) δl δz tt B 1 (13) T 2 = ξl + W δl δz x + D t (W ) δl δz tx + D x (W ) δl δz xx B 2 (14) is a conserved vector for Equation (10) associated with the operator X, where W = η z t τ z x ξ is the Lie characteristic function. 3. Conservation Laws of Equation (4) Consider the Benjamin Bona Mahony Equation (4). Here, we note that Equation (4) does not admit a Lagrangian. Nevertheless, we can transform Equation (4) into a variational form by setting u = z x. Thus, Equation (4) becomes a fourth-order equation, namely z tx + α(t)z xx + β(t)z x z xx + δz xxxt = 0 (15) It can easily be verified that a second-order Lagrangian of the Equation (15) is L = 1 { δz xx z xt α(t)zx 2 z t z x 1 } 2 3 β(t)z3 x (16) The substitution of L from Equation (16) into Equation (12) and splitting with respect to the derivatives of z, yields an overdetermined system of linear PDEs τ z = 0, η z = 0, ξ z = 0, τ x = 0, ξ t = 0, ξ x = 0, (17) η xx = 0, η xt = 0, β (t)τ + β(t)τ t = 0, (18) α (t)ξ 1 + β(t)η x + α(t)τ t = 0, (19) 1 2 η t α(t)η x = B 2 z, 1 2 η x = B 1 z, (20) Bt 1 + Bx 2 = 0. (21) After some lengthy calculations, the solution of the above system yields τ = a(t) (22) ξ = c 1 (23) η = c 2 x + d(t) (24) B 1 = 1 2 c 2z + f(t, x) (25) B 2 = 1 2 d z α(t)c 2 z + e(t, x) (26) f t + e x = 0 (27)
5 Symmetry 2014, β (t)a(t) + β(t)a (t) = 0 (28) α (t)a(t) + α(t)a (t) + β(t)c 2 = 0 (29) The analysis of Equations (28) and (29) prompts the following four cases: Case 1. α(t) and β(t) arbitrary but not of the form contained in Cases 2 4. In this case, we obtain two Noether point symmetries. These are given below together with their corresponding gauge functions: X 1 = x, B1 = f, B 2 = e, f t + e x = 0 (30) X 2 = d(t) z, B1 = f, B 2 = 1 2 d (t)z + e, f t + e x = 0 (31) Invoking Theorem 1, and reverting back to the original variables, the two nontrivial conserved vectors associated with these two Noether point symmetries are, respectively, and T 1 1 = u2 2 + δ 2 uu xx δ 2 u2 x f (32) T 2 1 = 1 2 α(t)u β(t)u3 + δuu xt δ 2 u xu t e (33) T2 1 = 1 2 d(t)u δ 2 d(t)u xx f (34) T2 2 = α(t)d(t)u 1 2 d(t) u t dx 1 2 β(t)d(t)u2 δd(t)u xt (35) δd (t)u x d (t) udx e Here it can be seen that the above conserved vectors do not satisfy the divergence condition, viz., D i T i (4) = 0, as some excessive terms emerge that require some further analysis. By making a slight adjustment to these terms, it can be shown that these terms can be absorbed into the divergence condition. For, hence D t (T 1 1 ) + D x (T 2 1 ) = δ 2 uu xxt δ 2 u xu xt f t e x = D t ( δ 2 uu xx f) D x ( δ 2 u xu t + e) (36) D t (T 1 1 δ 2 uu xx + f) + D x (T δ 2 u xu t + e) = 0. (37) We now redefine the conserved vectors in the parenthesis as: T 1 1 = T 1 1 δ 2 uu xx + f = 1 2 u2 δ 2 u2 x (38) T 2 1 = T δ 2 u xu t + e = α(t) 2 u2 + β(t) 3 u3 + δuu xt (39)
6 Symmetry 2014, Thus, the modified conserved vectors T 1 1 and T 2 1 satisfy the divergence condition. We observe that the conserved vectors Equations (38) and (39) are local conserved vectors. Likewise, we can then construct the second pair of the conserved quantities associated with T 1 2 and T 2 2 as: 2 1 = d(t) 2 u δ 2 d(t)u xx (40) 2 2 = α(t)d(t)u 1 2 d(t) u t dx β(t) 2 d(t)u2 δ 2 d(t)u xt + δ 2 d (t)u x d (t) udx (41) Note that the conserved quantities Equations (40) and (41) are nonlocal conserved vectors, and since d(t) is an arbitrary function of t, one obtains infinitely many nonlocal conserved vectors of Equation (4). A special case of Equations (40) and (41), when d(t) = 1, is 2 1 = u 2 δ 2 u xx (42) 2 2 = α(t)u 1 u t dx β(t) 2 2 u2 δ 2 u xt (43) Case 2. α(t) = γ, β(t) = λ, where γ and λ are non-zero constants. This case provides us with three Noether symmetry generators, namely, X 1, X 2 given by the operators Equations (30) and (31) and X 3 given by X 3 = t with B1 = f, B 2 = e, f t + e x = 0 (44) The use of Noether conserved vector Equations (13) and (14) corresponding to X 3 yields T3 1 = 1 2 γu2 1 6 λu3 + δ 2 u xx u t dx f (45) T3 2 = γu u t dx + 1 [ ] 2 u t dx λu2 u t dx + δu xt u t dx 1 2 δu x u tt dx 1 2 δu2 t e (46) Again, the above conserved flow fails to satisfy the divergence criterion. Thus, by inheriting the same procedure as in Case 1, the nontrivial conserved flows associated with X 1, X 2 and X 3 are T 1 1 = 1 2 u2 δ 2 u2 x (47) T 2 1 = γ 2 u2 + λ 3 u3 + δuu xt (48) 2 1 = d(t) 2 u δ 2 d(t)u xx (49) 2 2 = γd(t)u 1 2 d(t) u t dx λ 2 d(t)u2 δ 2 d(t)u xt + δ 2 d (t)u x d (t) udx (50)
7 Symmetry 2014, and 3 1 = 1 2 γu2 1 6 λu3 (51) 3 2 = γu u t dx + 1 [ ] 2 u t dx λu2 u t dx + δu xt u t dx 1 2 δu2 t (52) respectively. The conserved vector Equations (47) and (48) is a local conserved vector whereas the conserved vectors Equations (49) and (52) are nonlocal conserved quantities. It should be noted that one can use Equations (49) and (50) to construct infinitely many nonlocal conserved vectors. Case 3. α(t) = γe σt, β(t) = λe σt, where γ, σ and λ are nonzero constants. Here we get three Noether point symmetries operators, viz., X 1, X 2 Equations (30) and (31) and X 3 given by given by the operators X 3 = e σt t with B1 = f, B 2 = e, f t + e x = 0 (53) The application of Theorem 1, with X 3, gives T3 1 = 1 2 γu2 1 6 λu3 + δ 2 e σt u xx u t dx f (54) T3 2 = γu u t dx + 1 [ ] 2 2 e σt u t dx λu2 u t dx + δe σt u xt u t dx 1 2 δe σt u x u tt dx δσe σt u x u t dx 1 2 δe σt u 2 t e (55) and as before, the modified conserved vectors are given by 3 1 = T3 1 δ 2 e σt u xx u t dx = γ 2 u2 λ 6 u3 3 2 = T δσe σt u x u t dx δe σt u x = γu 1 2 δe σt u 2 t u t dx e σt [ ] 2 u t dx λu2 u tt dx u t dx + δe σt u xt Thus, the nontrivial conserved flows associated with X 1, X 2 and X 3 are, respectively, u t dx (56) (57) T 1 1 = 1 2 u2 δ 2 u2 x (58) T 2 1 = γ 2 eσt u 2 + λ 3 eσt u 3 + δuu xt (59) T 1 2 = 1 2 d(t)u 1 2 δd(t)u xx (60)
8 Symmetry 2014, = γe σt d(t)u 1 2 d(t) + δ 2 d (t)u x d (t) u t dx λ 2 eσt d(t)u 2 δ 2 d(t)u xt udx (61) 3 1 = γ 2 u2 λ 6 u3 3 2 = γu u t dx + 1 [ 2 e σt ] 2 u t dx λu2 u t dx + δe σt u xt u t dx (62) 1 2 δe σt u 2 t (63) Case 4. α(t) = γ/a(t), β(t) = λ/a(t), where γ, λ are constants, with γ, λ 0 and a(t) an arbitrary function of t. In this case, we obtain three Noether point symmetry generators, viz., X 1, X 2 given by Equations (30) and (31) and X 3 given by X 3 = a(t) t with B1 = f, B 2 = e, f t + e x = 0 (64) Following the above procedure, the nontrivial local and nonlocal conserved quantities corresponding to X 1, X 2 and X 3, in this case, are T 1 1 = δ 2 u2 δ 2 u2 x (65) T 2 1 = γ 2 a(t)u2 + λ 3 a(t)u3 + δuu xt (66) 2 1 = d(t) 2 2 = γ 2 u δ 2 d(t)u xx (67) a(t) d(t)u 1 2 d(t) u t dx λ a(t) d(t)u2 δ 2 d(t)u xt + δ 2 d (t)u x d (t) udx (68) 3 1 = 1 2 γu2 1 6 λu3 (69) 3 2 = γu u t dx + 1 [ ] 2 2 a(t) u t dx λu2 u t dx + δa(t)u xt u t dx 1 2 δa(t)u2 t (70) respectively. Remark 1. Remark: It should be noted that since the Lagrangian Equation (16) is invariant under the spatial translation symmetry, this will give rise to the linear momentum conservation laws. Moreover, if α(t) = γ, β(t) = λ, where γ and λ are non-zero constants, then the corresponding Lagrangian Equation (16) is also invariant under the time translation symmetry and thus the linear momentum and energy are both conserved.
9 Symmetry 2014, Exact Solution of Equation (4) for a Special Case Using Conservation Laws First we recall a definition and theorem from [15] that we utilize in this Section. Definition 2. Suppose that X is a symmetry of Equation (10) and T a conserved vector of Equation (10). Then if X and T satisfy then X is said to be associated with T. X(T i ) + T i D j (ξ j ) T j D j (ξ i ) = 0, i = 1, 2 (71) Define a nonlocal variable v by T t = v x, T x = v t. Then using the similarity variables r, s, w with the generator X = s, we have T r = v s, T s = v r, and the conservation law is then rewritten as where D r T r + D s T s = 0, T r = T s = T t D t (r) + T x D x (r) D t (r)d x (s) D x (r)d t (s) T t D t (s) + T x D x (s) D t (r)d x (s) D x (r)d t (s) (72) (73) Theorem 2. An n-th order partial differential equation with two independent variables, which admits a symmetry X that is associated with a conserved vector T, is reduced to an ordinary differential equation of order n 1; namely, T r = k, where T r is defined by Equation (72) for solutions invariant under X. We now use the above definition and theorem to obtain an exact solution of one special case of Equation (4) by making use of its conservation law. We consider Case 2 of Section 3. Recall that the Equation (4) with α(t) = γ, β(t) = λ, where γ and λ are non-zero constants, admits (among others) X 1 = t, X 2 = x and possesses the conservation law with conserved vector ( 1 T = 2 u2 δ γ 2 u2 x, 2 u2 + λ ) 3 u3 + δuu xt (74) (75) that is associated with both X 1 and X 2. We now define the combination of X 1 and X 2 by X = ρx 1 +X 2. Thus, the canonical coordinates of X are given by s = x, r = ρx t, u = u(r) (76) where u = u(r) is the invariant solution under X if u satisfies Equation (4) with α(t) = γ, β(t) = λ. Employing Equation (72), the r-component of T given in Equation (75) is T r = 1 2 u2 αγ 2 u2 δα2 2 u2 r αλ 3 u3 + δα 2 uu rr (77)
10 Symmetry 2014, Using Theorem 2 above, Equation (77) can be written as 1 2 u2 αγ 2 u2 δα2 2 u2 r αλ 3 u3 + δα 2 uu rr = k (78) By letting u r = p(u), we have u rr = p dp. Then Equation (78) reduces to the first order ordinary du differential equation p 1 2u p = 1 ( λ p 3δα u2 1 2δα u + 2 γ 2δα u + k ) δα 2 u The integration of Equation (79) leads to the four parameter family of solutions ± (λu 3 /3δα 2k/α 2 δ u 2 /α 2 δ + γu 2 /δα + k 1 u 2 ) 1 2 du = ρx t + k2 (80) of Equation (4) invariant under X = ρx 1 + X 2. (79) 5. Concluding Remarks In this paper we derived the conservation laws for the Benjamin Bona Mahony equation with variable coefficients Equation (4). This equation does not have a Lagrangian. In order to use Noether theorem, we transformed Equation (4) to a fourth-order Equation (15) which admits a Lagrangian. We then employed the Noether theorem to derive the conservation laws for Equation (15). The derived conserved vectors needed further adjustment to satisfy the divergence criterion. By reverting back to the original variable u, we constructed the conserved vectors for our third-order Benjamin Bona Mahony equation with variable coefficients Equation (4). The conserved vectors consisted of some local and infinite number of nonlocal conserved vectors. Finally, we obtained an exact solution for the Benjamin Bona Mahony equation using the double reduction theory. The importance of finding conservation laws and their physical applications are mentioned in the paper. Acknowledgments The authors thank the anonymous referees whose comments helped to improve the paper. Ben Muatjetjeja would like to thank the North-West University, Mafikeng Campus for their financial support. Author Contributions Ben Muatjetjeja and Chaudry Masood Khalique worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript. Conflicts of Interest The authors declare no conflict of interest.
11 Symmetry 2014, References 1. Benjamin, T.B.; Bona, J.L.; Mahony, J.J. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. A 1972, 272, Singh, K.; Gupta, R.K.; Kumar, S. Benjamin Bona Mahony (BBM) equation with variable coefficients: Similarity reductions and Painlevé analysis. Appl. Math. Comput. 2011, 217, Peregrine, D.H. Calculations of the development of an undular bore. J. Fluid Mech. 1966, 25, Molati, M.; Khalique, C.M. Lie symmetry analysis of the time-variable coefficient B-BBM equation. Adv. Differ. Equ. 2012, 212, doi: / Johnpillai, A.G.; Khalique, C.M. Variational approaches to conservation laws for a nonlinear evolution equation with time dependent coefficients. Quaest. Math. 2011, 34, Wang, M.L.; Wang, Y.M. A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients. Phys. Lett. A 2001, 287, Engui, F. Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations. Phys. Lett. A 2002, 294, Singh, K.; Gupta, R.K. Lie symmetries and exact solutions of a new generalized Hirota-Satsuma coupled KdV system with variable coefficients. Int. J. Eng. Sci. 2006, 44, Olver, P.J. Applications of Lie Groups to Differential Equations; Springer-Verlag: New York, NY, USA, Lax, P.D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 1968, 21, Benjamin, T.B. The stability of solitary waves. Proc. R. Soc. Lond. A 1972, 328, Knops, R.J.; Stuart, C.A. Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Ration. Mech. Anal. 1984, 86, LeVeque, R.J. Numerical Methods for Conservation Laws; Birkhäuser: Basel, Switzerland, Godlewski, E.; Raviart, P.A. Numerical Approximation of Hyperbolic Systems of Conservation laws; Springer: Berlin, Germany, Sjöberg, A. Double reduction of PDEs from the association of symmetries with conservation laws with applications. Appl. Math. Comput. 2007, 184, Muatjetjeja, B.; Khalique, C.M. Lie group classification for a generalized coupled Lane-Emden system in dimension one. East Asian J. Appl. Math. 2014, 4, Noether, E. Invariante Variationsprobleme. König. Gesell. Wissen. Göttingen. Math Phys. Klasse 1918, 2, Ibragimov, N.H. CRC Handbook of Lie Group Analysis of Differential Equations; CRC Press: Boca Raton, FL, USA, ; Volumes 1 3. c 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (
Research Article Exact Solutions of φ 4 Equation Using Lie Symmetry Approach along with the Simplest Equation and Exp-Function Methods
Abstract and Applied Analysis Volume 2012, Article ID 350287, 7 pages doi:10.1155/2012/350287 Research Article Exact Solutions of φ 4 Equation Using Lie Symmetry Approach along with the Simplest Equation
More informationPainlevé analysis and some solutions of variable coefficient Benny equation
PRAMANA c Indian Academy of Sciences Vol. 85, No. 6 journal of December 015 physics pp. 1111 11 Painlevé analysis and some solutions of variable coefficient Benny equation RAJEEV KUMAR 1,, R K GUPTA and
More informationGroup analysis, nonlinear self-adjointness, conservation laws, and soliton solutions for the mkdv systems
ISSN 139-5113 Nonlinear Analysis: Modelling Control, 017, Vol., No. 3, 334 346 https://doi.org/10.15388/na.017.3.4 Group analysis, nonlinear self-adjointness, conservation laws, soliton solutions for the
More informationAlexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011
Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan,
More informationSymmetry Reductions of (2+1) dimensional Equal Width. Wave Equation
Authors: Symmetry Reductions of (2+1) dimensional Equal Width 1. Dr. S. Padmasekaran Wave Equation Asst. Professor, Department of Mathematics Periyar University, Salem 2. M.G. RANI Periyar University,
More informationResearch Article Equivalent Lagrangians: Generalization, Transformation Maps, and Applications
Journal of Applied Mathematics Volume 01, Article ID 86048, 19 pages doi:10.1155/01/86048 Research Article Equivalent Lagrangians: Generalization, Transformation Maps, and Applications N. Wilson and A.
More informationPainlevé Analysis, Lie Symmetries and Exact Solutions for Variable Coefficients Benjamin Bona Mahony Burger (BBMB) Equation
Commun. Theor. Phys. 60 (2013) 175 182 Vol. 60, No. 2, August 15, 2013 Painlevé Analysis, Lie Symmetries and Exact Solutions for Variable Coefficients Benjamin Bona Mahony Burger (BBMB) Equation Vikas
More informationConservation laws for the geodesic equations of the canonical connection on Lie groups in dimensions two and three
Appl Math Inf Sci 7 No 1 311-318 (013) 311 Applied Mathematics & Information Sciences An International Journal Conservation laws for the geodesic equations of the canonical connection on Lie groups in
More informationConservation laws for Kawahara equations
Conservation laws for Kawahara equations Igor L. Freire Júlio C. S. Sampaio Centro de Matemática, Computação e Cognição, CMCC, UFABC 09210-170, Santo André, SP E-mail: igor.freire@ufabc.edu.br, julio.sampaio@ufabc.edu.br
More informationOn the Linearization of Second-Order Dif ferential and Dif ference Equations
Symmetry, Integrability and Geometry: Methods and Applications Vol. (006), Paper 065, 15 pages On the Linearization of Second-Order Dif ferential and Dif ference Equations Vladimir DORODNITSYN Keldysh
More information-Expansion Method For Generalized Fifth Order KdV Equation with Time-Dependent Coefficients
Math. Sci. Lett. 3 No. 3 55-6 04 55 Mathematical Sciences Letters An International Journal http://dx.doi.org/0.785/msl/03039 eneralized -Expansion Method For eneralized Fifth Order KdV Equation with Time-Dependent
More information2. The generalized Benjamin- Bona-Mahony (BBM) equation with variable coefficients [30]
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.1,pp.95-99 The Modified Sine-Cosine Method and Its Applications to the Generalized K(n,n) and BBM Equations
More informationBreaking soliton equations and negative-order breaking soliton equations of typical and higher orders
Pramana J. Phys. (2016) 87: 68 DOI 10.1007/s12043-016-1273-z c Indian Academy of Sciences Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders ABDUL-MAJID
More informationSoliton Solutions of a General Rosenau-Kawahara-RLW Equation
Soliton Solutions of a General Rosenau-Kawahara-RLW Equation Jin-ming Zuo 1 1 School of Science, Shandong University of Technology, Zibo 255049, PR China Journal of Mathematics Research; Vol. 7, No. 2;
More informationTravelling wave solutions for a CBS equation in dimensions
AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 Travelling wave solutions for a CBS equation in + dimensions MARIA LUZ GANDARIAS University of Cádiz Department
More informationNew solutions for a generalized Benjamin-Bona-Mahony-Burgers equation
AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 New solutions for a generalized Benjamin-Bona-Mahony-Burgers equation MARIA S. BRUZÓN University of Cádiz Department
More informationConservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations.
Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Alexey Shevyakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon,
More informationSymmetries and reduction techniques for dissipative models
Symmetries and reduction techniques for dissipative models M. Ruggieri and A. Valenti Dipartimento di Matematica e Informatica Università di Catania viale A. Doria 6, 95125 Catania, Italy Fourth Workshop
More informationDerivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations
Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations H. A. Erbay Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794,
More informationGalerkin method for the numerical solution of the RLW equation using quintic B-splines
Journal of Computational and Applied Mathematics 19 (26) 532 547 www.elsevier.com/locate/cam Galerkin method for the numerical solution of the RLW equation using quintic B-splines İdris Dağ a,, Bülent
More informationNew conservation laws for inviscid Burgers equation
Volume 31, N. 3, pp. 559 567, 2012 Copyright 2012 SBMAC ISSN 0101-8205 / ISSN 1807-0302 (Online) www.scielo.br/cam New conservation laws for inviscid Burgers equation IGOR LEITE FREIRE Centro de Matemática,
More informationExact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized ( G. )-Expansion Method
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Published by Canadian Center of Science and Education Exact Solutions for the Nonlinear +-Dimensional Davey-Stewartson Equation
More informationExact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially Integrable Equations
Thai Journal of Mathematics Volume 5(2007) Number 2 : 273 279 www.math.science.cmu.ac.th/thaijournal Exact Solutions of The Regularized Long-Wave Equation: The Hirota Direct Method Approach to Partially
More informationNew explicit solitary wave solutions for (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation
Physics Letters A 07 (00) 107 11 www.elsevier.com/locate/pla New explicit solitary wave solutions for ( + 1)-dimensional Boussinesq equation and ( + 1)-dimensional KP equation Yong Chen, Zhenya Yan, Honging
More informationMaejo International Journal of Science and Technology
Full Paper Maejo International Journal of Science and Technology ISSN 905-7873 Available online at www.mijst.mju.ac.th New eact travelling wave solutions of generalised sinh- ordon and ( + )-dimensional
More informationSymmetries and Group Invariant Reductions of Integrable Partial Difference Equations
Proceedings of 0th International Conference in MOdern GRoup ANalysis 2005, 222 230 Symmetries and Group Invariant Reductions of Integrable Partial Difference Equations A. TONGAS, D. TSOUBELIS and V. PAPAGEORGIOU
More informationResearch Article Some Results on Equivalence Groups
Applied Mathematics Volume 2012, Article ID 484805, 11 pages doi:10.1155/2012/484805 Research Article Some Results on Equivalence Groups J. C. Ndogmo School of Mathematics, University of the Witwatersrand,
More informationSymmetry reductions for the Tzitzeica curve equation
Fayetteville State University DigitalCommons@Fayetteville State University Math and Computer Science Working Papers College of Arts and Sciences 6-29-2012 Symmetry reductions for the Tzitzeica curve equation
More informationResearch Article Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 520491, 7 pages http://dx.doi.org/10.1155/2015/520491 Research Article Dynamic Euler-Bernoulli Beam Equation:
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 1, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 1, 15-22 ISSN: 1927-5307 BRIGHT AND DARK SOLITON SOLUTIONS TO THE OSTROVSKY-BENJAMIN-BONA-MAHONY (OS-BBM) EQUATION MARWAN ALQURAN
More informationExistence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy
Entropy 215, 17, 3172-3181; doi:1.339/e1753172 OPEN ACCESS entropy ISSN 199-43 www.mdpi.com/journal/entropy Article Existence of Ulam Stability for Iterative Fractional Differential Equations Based on
More informationExact solutions through symmetry reductions for a new integrable equation
Exact solutions through symmetry reductions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX, 1151 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More informationB.7 Lie Groups and Differential Equations
96 B.7. LIE GROUPS AND DIFFERENTIAL EQUATIONS B.7 Lie Groups and Differential Equations Peter J. Olver in Minneapolis, MN (U.S.A.) mailto:olver@ima.umn.edu The applications of Lie groups to solve differential
More informationA NONLINEAR GENERALIZATION OF THE CAMASSA-HOLM EQUATION WITH PEAKON SOLUTIONS
A NONLINEAR GENERALIZATION OF THE CAMASSA-HOLM EQUATION WITH PEAKON SOLUTIONS STEPHEN C. ANCO 1, ELENA RECIO 1,2, MARÍA L. GANDARIAS2, MARÍA S. BRUZÓN2 1 department of mathematics and statistics brock
More informationEvolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures
Symmetry, Integrability and Geometry: Methods and Applications Evolutionary Hirota Type +-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures Mikhail B. SHEFTEL and Devrim
More informationSUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO S 5TH-ORDER MKDV EQUATION
Journal of Applied Analysis and Computation Volume 7, Number 4, November 07, 47 430 Website:http://jaac-online.com/ DOI:0.94/0706 SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO S 5TH-ORDER MKDV EQUATION
More informationTraveling Wave Solutions For The Fifth-Order Kdv Equation And The BBM Equation By ( G G
Traveling Wave Solutions For The Fifth-Order Kdv Equation And The BBM Equation By ( )-expansion method Qinghua Feng Shandong University of Technology School of Science Zhangzhou Road 1, Zibo, 55049 China
More informationOn Symmetry Group Properties of the Benney Equations
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 1, 214 218 On Symmetry Group Properties of the Benney Equations Teoman ÖZER Istanbul Technical University, Faculty of Civil
More informationON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS
Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific (2007 ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS ANASTASIOS
More informationSoliton solutions of some nonlinear evolution equations with time-dependent coefficients
PRAMANA c Indian Academy of Sciences Vol. 80, No. 2 journal of February 2013 physics pp. 361 367 Soliton solutions of some nonlinear evolution equations with time-dependent coefficients HITENDER KUMAR,
More informationSYMMETRY REDUCTION AND NUMERICAL SOLUTION OF A THIRD-ORDER ODE FROM THIN FILM FLOW
Mathematical and Computational Applications,Vol. 15, No. 4, pp. 709-719, 2010. c Association for Scientific Research SYMMETRY REDUCTION AND NUMERICAL SOLUTION OF A THIRD-ORDER ODE FROM THIN FILM FLOW E.
More informationInternational Journal of Modern Mathematical Sciences, 2012, 3(2): International Journal of Modern Mathematical Sciences
Article International Journal of Modern Mathematical Sciences 2012 3(2): 63-76 International Journal of Modern Mathematical Sciences Journal homepage:wwwmodernscientificpresscom/journals/ijmmsaspx On Goursat
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More informationarxiv: v2 [nlin.si] 23 Sep 2016
Enhanced group classification of Benjamin Bona Mahony Burgers equations Olena Vaneeva 1, Severin Pošta 2 and Christodoulos Sophocleous 3 Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str.,
More informationFORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY
Jrl Syst Sci & Complexity (2007) 20: 284 292 FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY Muhammad USMAN Bingyu ZHANG Received: 14 January 2007 Abstract It
More informationBäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations
arxiv:nlin/0102001v1 [nlin.si] 1 Feb 2001 Bäcklund transformation and special solutions for Drinfeld Sokolov Satsuma Hirota system of coupled equations Ayşe Karasu (Kalkanli) and S Yu Sakovich Department
More informationA MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE
Pacific Journal of Applied Mathematics Volume 1, Number 2, pp. 69 75 ISSN PJAM c 2008 Nova Science Publishers, Inc. A MULTI-COMPONENT LAX INTEGRABLE HIERARCHY WITH HAMILTONIAN STRUCTURE Wen-Xiu Ma Department
More informationKink, singular soliton and periodic solutions to class of nonlinear equations
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10 Issue 1 (June 015 pp. 1 - Applications and Applied Mathematics: An International Journal (AAM Kink singular soliton and periodic
More informationThe Traveling Wave Solutions for Nonlinear Partial Differential Equations Using the ( G. )-expansion Method
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.8(009) No.4,pp.435-447 The Traveling Wave Solutions for Nonlinear Partial Differential Equations Using the ( )-expansion
More informationTravelling Wave Solutions and Conservation Laws of Fisher-Kolmogorov Equation
Gen. Math. Notes, Vol. 18, No. 2, October, 2013, pp. 16-25 ISSN 2219-7184; Copyright c ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in Travelling Wave Solutions and
More informationSymmetry reductions and exact solutions for the Vakhnenko equation
XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, 1-5 septiembre 009 (pp. 1 6) Symmetry reductions and exact solutions for the Vakhnenko equation M.L.
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationNUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT
ANZIAM J. 44(2002), 95 102 NUMERICAL SOLITARY WAVE INTERACTION: THE ORDER OF THE INELASTIC EFFECT T. R. MARCHANT 1 (Received 4 April, 2000) Abstract Solitary wave interaction is examined using an extended
More informationEXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT EQUATION
Journal of Applied Analysis and Computation Volume 5, Number 3, August 015, 485 495 Website:http://jaac-online.com/ doi:10.11948/015039 EXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT
More informationNew Approach of ( Ǵ/G ) Expansion Method. Applications to KdV Equation
Journal of Mathematics Research; Vol. 6, No. ; ISSN 96-9795 E-ISSN 96-989 Published by Canadian Center of Science and Education New Approach of Ǵ/G Expansion Method. Applications to KdV Equation Mohammad
More informationB-splines Collocation Algorithms for Solving Numerically the MRLW Equation
ISSN 1749-889 (print), 1749-897 (online) International Journal of Nonlinear Science Vol.8(2009) No.2,pp.11-140 B-splines Collocation Algorithms for Solving Numerically the MRLW Equation Saleh M. Hassan,
More informationSOLITON SOLUTIONS OF SHALLOW WATER WAVE EQUATIONS BY MEANS OF G /G EXPANSION METHOD
Journal of Applied Analysis and Computation Website:http://jaac-online.com/ Volume 4, Number 3, August 014 pp. 1 9 SOLITON SOLUTIONS OF SHALLOW WATER WAVE EQUATIONS BY MEANS OF G /G EXPANSION METHOD Marwan
More informationSymmetry Methods for Differential Equations and Conservation Laws. Peter J. Olver University of Minnesota
Symmetry Methods for Differential Equations and Conservation Laws Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver Santiago, November, 2010 Symmetry Groups of Differential Equations
More informationGalilean invariance and the conservative difference schemes for scalar laws
RESEARCH Open Access Galilean invariance and the conservative difference schemes for scalar laws Zheng Ran Correspondence: zran@staff.shu. edu.cn Shanghai Institute of Applied Mathematics and Mechanics,
More informationNew Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation
International Conference on Computer Technology and Science (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V47.59 New Analytical Solutions For () Dimensional Kaup-Kupershmidt Equation
More informationIntroduction to the Hirota bilinear method
Introduction to the Hirota bilinear method arxiv:solv-int/9708006v1 14 Aug 1997 J. Hietarinta Department of Physics, University of Turku FIN-20014 Turku, Finland e-mail: hietarin@utu.fi Abstract We give
More informationON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL EQUATION
Journal of Nonlinear Mathematical Physics ISSN: 1402-9251 (Print) 1776-0852 (Online) Journal homepage: http://www.tandfonline.com/loi/tnmp20 ON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL
More informationLie Symmetry of Ito Stochastic Differential Equation Driven by Poisson Process
American Review of Mathematics Statistics June 016, Vol. 4, No. 1, pp. 17-30 ISSN: 374-348 (Print), 374-356 (Online) Copyright The Author(s).All Rights Reserved. Published by American Research Institute
More informationANALYSIS OF A NONLINEAR SURFACE WIND WAVES MODEL VIA LIE GROUP METHOD
Electronic Journal of Differential Equations, Vol. 206 (206), No. 228, pp. 8. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS OF A NONLINEAR SURFACE WIND WAVES MODEL
More informationResearch Article Soliton Solutions for the Wick-Type Stochastic KP Equation
Abstract and Applied Analysis Volume 212, Article ID 327682, 9 pages doi:1.1155/212/327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Y. F. Guo, 1, 2 L. M. Ling, 2 and
More informationSymmetry reductions and travelling wave solutions for a new integrable equation
Symmetry reductions and travelling wave solutions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX 0, 50 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More informationSymbolic Computation and New Soliton-Like Solutions of the 1+2D Calogero-Bogoyavlenskii-Schif Equation
MM Research Preprints, 85 93 MMRC, AMSS, Academia Sinica, Beijing No., December 003 85 Symbolic Computation and New Soliton-Like Solutions of the 1+D Calogero-Bogoyavlenskii-Schif Equation Zhenya Yan Key
More informationInvariant and Conditionally Invariant Solutions of Magnetohydrodynamic Equations in (3 + 1) Dimensions
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 1, 118 124 Invariant and Conditionally Invariant Solutions of Magnetohydrodynamic Equations in 3 + 1) Dimensions A.M. GRUNDLAND
More informationJournal of Mathematical Analysis and Applications. Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations
J. Math. Anal. Appl. 357 (2009) 225 231 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Complete symmetry group and nonlocal symmetries
More informationApplications of Lie Group Analysis to the Equations of Motion of Inclined Unsagged Cables
Applied Mathematical Sciences, Vol., 008, no. 46, 59-69 Applications of Lie Group Analysis to the Equations of Motion of Inclined Unsagged Cables Waraporn Chatanin Department of Mathematics King Mongkut
More informationNatural States and Symmetry Properties of. Two-Dimensional Ciarlet-Mooney-Rivlin. Nonlinear Constitutive Models
Natural States and Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Nonlinear Constitutive Models Alexei Cheviakov, Department of Mathematics and Statistics, Univ. Saskatchewan, Canada Jean-François
More informationConditional linearization of the quintic nonlinear beam equation
Sripana and Chatanin Advances in Difference Equations 207) 207:9 DOI 0.6/s3662-07-79- R E S E A R C H Open Access Conditional linearization of the quintic nonlinear beam equation Nuntawan Sripana and Waraporn
More informationarxiv: v1 [math-ph] 25 Jul Preliminaries
arxiv:1107.4877v1 [math-ph] 25 Jul 2011 Nonlinear self-adjointness and conservation laws Nail H. Ibragimov Department of Mathematics and Science, Blekinge Institute of Technology, 371 79 Karlskrona, Sweden
More informationApplication of fractional sub-equation method to the space-time fractional differential equations
Int. J. Adv. Appl. Math. and Mech. 4(3) (017) 1 6 (ISSN: 347-59) Journal homepage: www.ijaamm.com IJAAMM International Journal of Advances in Applied Mathematics and Mechanics Application of fractional
More informationModified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics
Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics Elsayed M. E. Zayed Mathematics department, Faculty of Science Zagazig University, Zagazig,
More informationGeneralized evolutionary equations and their invariant solutions
Generalized evolutionary equations and their invariant solutions Rodica Cimpoiasu, Radu Constantinescu University of Craiova, 13 A.I.Cuza Street, 200585 Craiova, Dolj, Romania Abstract The paper appplies
More informationNew approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations
Volume 28, N. 1, pp. 1 14, 2009 Copyright 2009 SBMAC ISSN 0101-8205 www.scielo.br/cam New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations HASSAN A. ZEDAN Mathematics
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 60 (00) 3088 3097 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Symmetry
More informationDirect construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications
Euro. Jnl of Applied Mathematics (2002), vol. 13, pp. 545 566. c 2002 Cambridge University Press DOI: 10.1017/S095679250100465X Printed in the United Kingdom 545 Direct construction method for conservation
More informationAn Application of Equivalence Transformations to Reaction Diffusion Equations
Symmetry 2015, 7, 1929-1944; doi:10.3390/sym7041929 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Article An Application of Equivalence Transformations to Reaction Diffusion Equations
More informationu = λ u 2 in Ω (1) u = 1 on Ω (2)
METHODS AND APPLICATIONS OF ANALYSIS. c 8 International Press Vol. 15, No. 3, pp. 37 34, September 8 4 SYMMETRY ANALYSIS OF A CANONICAL MEMS MODEL J. REGAN BECKHAM AND JOHN A. PELESKO Abstract. A canonical
More informationApplication of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics
PRMN c Indian cademy of Sciences Vol. 77, No. 6 journal of December 011 physics pp. 103 109 pplication of the trial equation method for solving some nonlinear evolution equations arising in mathematical
More informationarxiv:nlin/ v1 [nlin.si] 25 Sep 2006
Remarks on the conserved densities of the Camassa-Holm equation Amitava Choudhuri 1, B. Talukdar 1a and S. Ghosh 1 Department of Physics, Visva-Bharati University, Santiniketan 73135, India Patha Bhavana,
More informationThe symmetry structures of ASD manifolds
The symmetry structures of ASD manifolds 1 1 School of Mathematics University of the Witatersrand, Johannesburg South Africa African Institute of Mathematics Outline 1 Introduction 2 3 The dispersionless
More informationA Recursion Formula for the Construction of Local Conservation Laws of Differential Equations
A Recursion Formula for the Construction of Local Conservation Laws of Differential Equations Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada December
More informationGeneralized bilinear differential equations
Generalized bilinear differential equations Wen-Xiu Ma Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Abstract We introduce a kind of bilinear differential
More informationThe (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics
Vol.3, Issue., Jan-Feb. 3 pp-369-376 ISSN: 49-6645 The ('/) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics J.F.Alzaidy Mathematics Department, Faculty
More informationThe General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics 015; 4(5): 335-341 Published online August 16 015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.0150405.11 ISSN: 38-5605 (Print); ISSN: 38-5613
More informationK. Ambika and R. Radha
Indian J. Pure Appl. Math., 473: 501-521, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0200-9 RIEMANN PROBLEM IN NON-IDEAL GAS DYNAMICS K. Ambika and R. Radha School of Mathematics
More informationSymmetry Classification of KdV-Type Nonlinear Evolution Equations
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 1, 125 130 Symmetry Classification of KdV-Type Nonlinear Evolution Equations Faruk GÜNGÖR, Victor LAHNO and Renat ZHDANOV Department
More informationGroup analysis of differential equations: A new type of Lie symmetries
Group analysis of differential equations: A new type of Lie symmetries Jacob Manale The Department of Mathematical Sciences, University of South Africa, Florida, 1709, Johannesburg, Gauteng Province, Republic
More informationarxiv: v1 [math.ap] 25 Aug 2009
Lie group analysis of Poisson s equation and optimal system of subalgebras for Lie algebra of 3 dimensional rigid motions arxiv:0908.3619v1 [math.ap] 25 Aug 2009 Abstract M. Nadjafikhah a, a School of
More informationConformable variational iteration method
NTMSCI 5, No. 1, 172-178 (217) 172 New Trends in Mathematical Sciences http://dx.doi.org/1.2852/ntmsci.217.135 Conformable variational iteration method Omer Acan 1,2 Omer Firat 3 Yildiray Keskin 1 Galip
More informationSome Soliton Solutions of Non Linear Partial Differential Equations by Tan-Cot Method
IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578,p-ISSN: 319-765X, 6, Issue 6 (May. - Jun. 013), PP 3-8 Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Cot Method Raj Kumar
More informationChebyshev Collocation Spectral Method for Solving the RLW Equation
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(2009) No.2,pp.131-142 Chebyshev Collocation Spectral Method for Solving the RLW Equation A. H. A. Ali Mathematics
More informationarxiv: v1 [nlin.si] 23 Aug 2007
arxiv:0708.3247v1 [nlin.si] 23 Aug 2007 A new integrable generalization of the Korteweg de Vries equation Ayşe Karasu-Kalkanlı 1), Atalay Karasu 2), Anton Sakovich 3), Sergei Sakovich 4), Refik Turhan
More informationA Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility
Commun. Theor. Phys. (Beijing, China) 52 (2009) pp. 398 402 c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 3, September 15, 2009 A Note on Nonclassical Symmetries of a Class of Nonlinear
More informationEXACT TRAVELLING WAVE SOLUTIONS FOR NONLINEAR SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS
Journal of Applied Analysis and Computation Volume 7, Number 4, November 2017, 1586 1597 Website:http://jaac-online.com/ DOI:10.11948/2017096 EXACT TRAVELLIN WAVE SOLUTIONS FOR NONLINEAR SCHRÖDINER EQUATION
More informationSimple waves and a characteristic decomposition of the two dimensional compressible Euler equations
Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations Jiequan Li 1 Department of Mathematics, Capital Normal University, Beijing, 100037 Tong Zhang Institute
More informationSYMMETRY ANALYSIS AND SOME SOLUTIONS OF GOWDY EQUATIONS
SYMMETRY ANALYSIS AND SOME SOLUTIONS OF GOWDY EQUATIONS RAJEEV KUMAR 1, R.K.GUPTA 2,a, S.S.BHATIA 2 1 Department of Mathematics Maharishi Markandeshwar Univesity, Mullana, Ambala-131001 Haryana, India
More information